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Recovering a moving boundary from Cauchy data in an inverse problem which arises in modeling brain tumor treatment: the (quasi)linearization idea combined with radial basis functions (RBFs) approximation

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Abstract

In this paper, the dynamical modeling and behavior analysis of the inverse boundary Stefan problem which promising the understanding of modeling brain tumor treatment, are studied. To umerical simulate these models and to overcome their difficulties such as, non-linearity, free boundary property and having a non-rectangular domain, we propose the use of a strongly meshless technique based on radial basis functions in conjunction with a (quasi)linearization algorithm. Numerical examples are given to show the good accuracy and stability of the presented method.

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References

  1. Beck JV, Blackwell B, Clair C (1985) Inverse conduction Ill-posed problems. Wiley, Amsterdam

    MATH  Google Scholar 

  2. Canon JR (1984) The one dimensional heat equation. Cambridge University Press, Cambridge

    Google Scholar 

  3. Johansson BT, Lesnic D, Reeve T (2011) A method of fundamental solutions for the one-dimensional inverse Stefan problem. Appl Math Model 35:4367–4378

    MathSciNet  MATH  Google Scholar 

  4. Johansson BT, Lesnic D, Reeve T (2011) Numerical approximation of the one-dimensional inverse Cauchy–Stefan problem using a method of fundamental solutions. Invese Probl Sci Eng 19:659–677

    MathSciNet  MATH  Google Scholar 

  5. Johansson BT, Lesnic D, Reeve T (2011) A method of fundamental solutions for two-dimensional heat conduction. Int J Comput Math 88:1697–1713

    MathSciNet  MATH  Google Scholar 

  6. Johansson BT, Lesnic D, Reeve T (2012) A method of fundamental solutions for radially symmetric and axisymmetric backward heat conduction problems. Int J Comput Math 89:1555–1568

    MathSciNet  MATH  Google Scholar 

  7. Johansson BT, Lesnic D, Reeve T (2012) A method of fundamental solutions for the radially symmetric inverse heat conduction problem. Int Commun Heat Mass 39:887–895

    MATH  Google Scholar 

  8. Jaroudi R, Baravdish G, Johansson BT, Astrom F (2019) Numerical reconstruction of brain tumours. Inverse Prob Sci Eng 27:278–298

    MathSciNet  MATH  Google Scholar 

  9. Chapkoa R, Johansson BT (2018) A boundary integral equation method for numerical solution of parabolic and hyperbolic cauchy problems. Appl Numer Math 129:104–119

    MathSciNet  Google Scholar 

  10. Huntul M, Lesnic D, Johansson BT (2018) Determination of an additive time- and space-dependent coefficient in the heat equation. Int J Numer Meth Heat Fluid Flow 28:1352–1373

    Google Scholar 

  11. Hadian AH, Rad JA (2019) A computational source modeling of brain activity: an inverse problem. J Neurodev Cognit 1:50–56

    Google Scholar 

  12. Chakraverty S (2020) Mathematical methods in interdisciplinary sciences. Wiley, Amsterdam

  13. Rashedi K, Adibi H, Rad JA, Parand K (2014) Application of the meshfree methods for solving the inverse one-dimensional Stefan problem. Eng Anal Bound Elem 40:1–21

    MathSciNet  MATH  Google Scholar 

  14. Sarabadan S, Rashedi K, Adibi H (2018) Boundary determination of the inverse heat conduction problem in one and two dimensions via the collocation method based on the satisfier functions. Iran J Sci Technol Trans Sci 42:827–840

    MathSciNet  MATH  Google Scholar 

  15. Gol’dman NL (1997) Inverse Stefan problems. Kluwer Academic Publ, Beijing

    MATH  Google Scholar 

  16. Lamé G, Clapeyron BP (1831) Mémoire sur la solidification par refroidissement d’un globe solide. Ann Chem Phys 47:250–256

    Google Scholar 

  17. Rubinsteın LI (1971) The Stefan problem. American Mathematical Society, New York

    Google Scholar 

  18. Stefan J (1889) Uber einige Probleme der Theorie der Wärmeleitung. S-B Wien Akad Mat Natur 98:473–484

    MATH  Google Scholar 

  19. Johansson BT, Lesnic D, Reeve T (2014) A method of fundamental solutions for the two-dimensional inverse Stefan problem. Inverse Probl Sci Eng 22:112–129

    MathSciNet  MATH  Google Scholar 

  20. Hsieh MH (2012) Mathematical modelling of controlled drug release from polymer micro-spheres: incorporating the effects of swelling, diffusion and dissolution via moving boundary problems. Ph.D. thesis, Queensland University of Technology

  21. Hsieh M, McCue SW, Moroney TJ, Nelson MI (2011) Drug diffusion from polymeric delivery devices: a problem with two moving boundaries. ANZIAM J 52:549–566

    MathSciNet  MATH  Google Scholar 

  22. Kipper MJ, Narasimhan B (2005) Molecular description of erosion phenomena in biodegradable polymers. Macromolecules 38:1989–1999

    Google Scholar 

  23. Faisant N, Akiki J, Siepmann F, Benoit JP, Siepmann J (2006) Effects of the type of release medium on drug release from PLGA-based microparticles: experiment and theory. I. J Pharmaceutics 314:189–197

    Google Scholar 

  24. Lao LL, Venkatraman SS, Peppas NA (2008) Modeling of drug release from biodegradable polymer blends. Eur J Pharm Biopharm 70:796–803

    Google Scholar 

  25. McCue SW, Hsieh M, Moroney TJ, Nelson MI (2011) Asymptotic and numerical results for a model of solvent-dependent drug diffusion through polymeric spheres. SIAM J Appl Math 71:2287–2311

    MathSciNet  MATH  Google Scholar 

  26. Wei T, Yamamoto M (2009) Reconstruction of a moving boundary from Cauchy data in one-dimensional heat equation. Inverse Probl Sci Eng 17:551–567

    MathSciNet  MATH  Google Scholar 

  27. Liu J, Guerrier B (1997) A comparative study of domain embedding method for regularized solutions of inverse Stefan problems. Int J Numer Methods Eng 40:3579–3600

    MathSciNet  MATH  Google Scholar 

  28. Hon YC, Li M (2008) A computational method for inverse free boundary determination problem. Int J Numer Methods Eng 73:1291–1309

    MathSciNet  MATH  Google Scholar 

  29. Fredman TP (2004) A boundary identification method for an inverse heat conduction problem with an application in ironmaking. Heat Mass Transf 41:95–103

    Google Scholar 

  30. Badia AE, Moutazaim F (1999) A one-phase inverse Stefan problem. Inverse Prob 15:1507–1522

    MathSciNet  MATH  Google Scholar 

  31. Larsson E, Shcherbakov V, Heryudono A (2017) A least squares radial basis function partition of unity method for solving PDEs. SIAM J Sci Comput 39:A2538–A2563

    MathSciNet  MATH  Google Scholar 

  32. Shcherbakov V, Larsson E (2016) Radial basis function partition of unity methods for pricing vanilla basket options. Comput Math Appl 71:185–200

    MathSciNet  MATH  Google Scholar 

  33. Safdari-Vaighani A, Heryudono A, Larsson E (2015) A radial basis function partition of unity collocation method for convection-diffusion equations arising in financial applications. J Sci Comput 64:341–367

    MathSciNet  MATH  Google Scholar 

  34. Shcherbakov V (2016) Radial basis function partition of unity operator splitting method for pricing multi-asset American options. BIT Numer Math 56:1401–1423

    MathSciNet  MATH  Google Scholar 

  35. Dehghan M, Ghesmati A (2010) The Meshless Local Petrov-Galerkin (MLPG) method: a simple and less-costly alternative to the finite element and boundary element methods. Eng Anal Bound Elem 34:324–336

    MathSciNet  MATH  Google Scholar 

  36. Dehghan M, Mirzaei D (2009) Meshless local boundary integral equation (LBIE) method for the unsteady magnetohydrodynamic (MHD) flow in rectangular and circular pipes. Comput Phys Commun 180:1458–1466

    MathSciNet  Google Scholar 

  37. Rad JA, Parand K, Ballestra LV Pricing European and American options by radial basis point interpolation. Appl Math Comput 251(201):363–377

  38. Assari P, Adibi H, Dehghan M (2013) A meshless method for solving nonlinear two-dimensional integral equations of the second kind on non-rectangular domains using radial basis functions with error analysis. J Comput Appl Math 239:72–92

    MathSciNet  MATH  Google Scholar 

  39. Assari P, Adibi H, Dehghan M (2014) A meshless discrete galerkin (MDG) method for the numerical solution of integral equations with logarithmic kernels. J Comput Appl Math 267:160–181

    MathSciNet  MATH  Google Scholar 

  40. Dehghan M, Najafi M (2016) Numerical solution of a non-classical two-phase Stefan problem via radial basis function (RBF) collocation methods. Eng Anal Bound Element 72:111–127

    MathSciNet  MATH  Google Scholar 

  41. Hemami M, Parand K, Rad JA (2019) Numerical simulation of reaction–diffusion neural dynamics models and their synchronization desynchronization: application to epileptic seizures. Comput Math Appl 78:3644–3677

    MathSciNet  MATH  Google Scholar 

  42. Shirzadi A, Ling L, Abbasbandy S (2012) Meshless simulations of the two-dimensional fractional-time convection–diffusion-reaction equations. Eng Anal Bound Elem 36:1522–1527

    MathSciNet  MATH  Google Scholar 

  43. Abbasbandy S, Shirzadi A (2011) MLPG method for two-dimensional diffusion equation with Neumann’s and non-classical boundary conditions. Appl Numer Math 61:170–180

    MathSciNet  MATH  Google Scholar 

  44. Abbasbandy S, Ghehsareh HR, Alhuthali MS, Alsulami HH (2014) Comparison of meshless local weak and strong forms based on particular solutions for a non-classical 2-D diffusion model. Eng Anal Bound Elem 39:121–128

    MathSciNet  MATH  Google Scholar 

  45. Dehghan M, Mirzaei D (2009) Meshless local Petrov-Galerkin (MLPG) method for the unsteady magnetohydrodynamic (MHD) flow through pipe with arbitrary wall conductivity. Appl Numer Math 59:1043–1058

    MathSciNet  MATH  Google Scholar 

  46. Dehghan M, Mirzaei D (2008) The meshless local Petrov-Galerkin MLPG method for the generalized two-dimensional non-linear Schrodinger equation. Eng Anal Bound Elem 32:747–756

    MATH  Google Scholar 

  47. Mohammadi V, Dehghan M (2018) Simulation of the phase field Cahn–Hilliard and tumor growth models via a numerical scheme: element-free Galerkin method. Comput Methods Appl Mech Eng in press

  48. Dehghan M, Narimani N (2018) An element-free Galerkin meshless method for simulating the behavior of cancer cell invasion of surrounding tissue. Appl Math Model 59:500–513

    MathSciNet  MATH  Google Scholar 

  49. Dehghan M, Abbaszadeh M (2016) Numerical study of three-dimensional Turing patterns using a meshless method based on moving Kriging element free Galerkin (EFG) approach. Comput Math Appl 72:427–454

    MathSciNet  MATH  Google Scholar 

  50. Dehghan M, Abbaszadeh M (2018) A combination of proper orthogonal decomposition-discrete empirical interpolation method (POD-DEIM) and meshless local RBF-DQ approach for prevention of groundwater contamination. Comput Math Appl 75:1390–1412

    MathSciNet  MATH  Google Scholar 

  51. Cheung K, Ling L (2018) A Kernel-based embedding method and convergence analysis for surfaces PDEs. SIAM J Sci Comput 40:A266–A287

    MathSciNet  MATH  Google Scholar 

  52. Petras A, Ling L, Ruuth S (2018) An RBF-FD closest point method for solving PDEs on surfaces. J Comput Phys 370:43–57

    MathSciNet  MATH  Google Scholar 

  53. Kansa EJ (1990) Multiquadrics-A scattered data approximation scheme with applications to computational fluid-dynamics-I surface approximations and partial derivative estimates. Comput Math Appl 19:127–145

    MathSciNet  MATH  Google Scholar 

  54. Kansa EJ (1990) Multiquadrics-A scattered data approximation scheme with applications to computational fluid-dynamics II. solutions to parabolic, hyperbolic and elliptic partial differential equations. Comput Math Appl 19:147–161

    MathSciNet  MATH  Google Scholar 

  55. Sharan M, Kansa EJ, Gupta S (1997) Application of the multiquadric method for numerical solution of elliptic partial differential equations. Appl Math Comput 84:275–302

    MathSciNet  MATH  Google Scholar 

  56. Milovanović S, Sydow L (2018) Radial basis function generated finite differences for option pricing problems. Comput Math Appl 75:1462–1481

    MathSciNet  MATH  Google Scholar 

  57. Jafarabadi A, Shivanian E (2018) Numerical simulation of nonlinear coupled Burgers’ equation through meshless radial point interpolation method. Eng Anal Bound Elem 95:187–199

    MathSciNet  MATH  Google Scholar 

  58. Shivanian E, Jafarabadi A (2018) The spectral meshless radial point interpolation method for solving an inverse source problem of the time-fractional diffusion equation. Appl Numer Math 129:1–25

    MathSciNet  MATH  Google Scholar 

  59. Shivanian E, Jafarabadi A (2018) Capillary formation in tumor angiogenesis through meshless weak and strong local radial point interpolation. Eng Comput 34:603–619

    Google Scholar 

  60. Shivanian E, Jafarabadi A (2018) An improved meshless algorithm for a kind of fractional cable problem with error estimate. Chaos Solit Fract 110:138–151

    MathSciNet  MATH  Google Scholar 

  61. Rad JA, Hook J, Larsson E, Sydow L (2018) Forward deterministic pricing of options using Gaussian radial basis functions. J Comput Sci 24:209–217

    MathSciNet  Google Scholar 

  62. Rad JA, Parand K (2017) Numerical pricing of American options under two stochastic factor models with jumps using a meshless local Petrov–Galerkin method. Appl Numer Math 115:252–274

    MathSciNet  MATH  Google Scholar 

  63. Rad JA, Parand K, Abbasbandy S (2015) Pricing European and American options using a very fast and accurate scheme-the meshless local Petrov–Galerkin (MLPG). Proc Natl Acad Sci India Sect A Phys Sci 85:337–351

    MATH  Google Scholar 

  64. Rad JA, Parand K, Abbasbandy S (2015) Local weak form meshless techniques based on the radial point interpolation (RPI) method and local boundary integral equation (LBIE) method to evaluate European and American options. Commun Nonlinear Sci Numer Simulat 22:1178–1200

    MathSciNet  MATH  Google Scholar 

  65. Rippa S (1999) An algorithm for selecting a good parameter \(c\) in radial basis function interpolation. Adv Comput Math 11:193–210

    MathSciNet  MATH  Google Scholar 

  66. Fornberg B, Wright G (2004) Stable computation of multiquadric interpolants for all values of the shape parameter. Comput Math Appl 48:853–867

    MathSciNet  MATH  Google Scholar 

  67. Landau HG (1950) Heat conduction in a melting solid. Q Appl Math 8:81–94

    MathSciNet  MATH  Google Scholar 

  68. Blackwell BF, Hogant RE (1994) One-dimensional ablation using Landau transformation and finite control volume procedure. J Thermophys Heat Transf 8:282–287

    Google Scholar 

  69. Parand K, Lotfi Y, Rad JA (2017) An accurate numerical analysis of the laminar two-dimensional flow of an incompressible Eyring–Powell fluid over a linear stretching sheet. Eur Phys J Plus 132:397

    Google Scholar 

  70. Parand K, Delkhosh M (2017) Accurate solution of the Thomas–Fermi equation using the fractional order of rational Chebyshev functions. J Comput Appl Math 317:624–642

    MathSciNet  MATH  Google Scholar 

  71. Mandelzweig VB, Tabakin F (2001) Quasilinearization approach to nonlinear problems in physics with application to nonlinear ODEs. Comput Phys Commun 141:268–281

    MathSciNet  MATH  Google Scholar 

  72. Ramos JI (2004) Piecewise quasilinearization techniques for singular boundary-value problems. Comput Phys Commun 158:12–25

    MathSciNet  MATH  Google Scholar 

  73. El-Gebeily M, O’Regan D (2006) A generalized quasilinearization method for second-order nonlinear differential equations with nonlinear boundary conditions. J Comput Appl Math 192:270–281

    MathSciNet  MATH  Google Scholar 

  74. Fasshauer G (2007) Meshfree approximation methods with MATLAB. Word Scientific Publishing, Singapore

    MATH  Google Scholar 

  75. Haq S, Islam S, Uddin M (2009) A mesh-free method for the numerical solution of the KdV-Burgers equation. Appl Math Model 33:3442–3449

    MathSciNet  MATH  Google Scholar 

  76. Mokhtari R, Mohammadi M (2010) Numerical solution of GRLW equation using Sinc-collocation method. Comput Phys Commun 181:1266–1274

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Computer Sciences, Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran, G.C., Iran

    A. Hajiollow, Y. Lotfi & K. Parand

  2. Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Canada

    K. Parand

  3. Department of Cognitive Modeling, Institute for Cognitive and Brain Sciences, Shahid Beheshti University, Tehran, G.C, Iran

    K. Parand, A. H. Hadian & J. A. Rad

  4. Department of Mathematics, University of Science and Technology of Mazandaran, Behshahr, Iran

    K. Rashedi

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  1. A. Hajiollow
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  2. Y. Lotfi
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  3. K. Parand
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  4. A. H. Hadian
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  5. K. Rashedi
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  6. J. A. Rad
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Hajiollow, A., Lotfi, Y., Parand, K. et al. Recovering a moving boundary from Cauchy data in an inverse problem which arises in modeling brain tumor treatment: the (quasi)linearization idea combined with radial basis functions (RBFs) approximation. Engineering with Computers 37, 1735–1749 (2021). https://doi.org/10.1007/s00366-019-00909-8

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